Fibonacci + Limericks?

I found some interesting (in my geeky sort of way) reading in a short essay, Self-similar syncopations: Fibonacci, L-systems, limericks and ragtime by Kevin Jones.

He draws some interesting parallels, but I feel that some of his observed relationships between the Fibonacci series, L-systems and limericks have more coincidence than anything else, particularly given the variation in the number of leading and trailing unstressed syllables in popular limericks. Kevin has chosen one particular form of limerick (iamb-anapest-anapest) as the archetypical limerick form and then sung its praises and noted the way it can be modeled with an L-system. His matching of the syllable counts to the Fibonacci series (which appear in so many other places in nature) seems to make sense of the world unless we look closer at his own selection process.

If he'd chosen one of the other very common limerick line forms (anapest-anapest-anapest, or amphibrach-amphibrach-amphibrach) his numbers wouldn't have worked out, and yet limericks written in those forms are just as catchy, just as "strangely appealing and intuitively 'natural'".

Let's take another example of an Edward Lear limerick.

There was a Young Lady whose bonnet,
Came untied when the birds sat upon it;
But she said, 'I don't care!
All the birds of the air
Make my Limerick seem like a Sonnet.

di dum di di dum di di dum di
di di dum di di dum di di dum di
di di dum di di dum
di di dum di di dum
di di dum di di dum di di dum di

This has 13 stressed syllables, 28 unstressed syllables (a total of 41), 9/10 syllables in the longer lines, and 6 in the shorter lines. By my count that's only 1 Fibonacci series number. Does that make the limerick sound unpalatable? Not at all! The deviation from perfect self-similarity in the lack of one unstressed syllable at the start of the first line doesn't hurt it at all. The change from 8 and 5 syllables per line to 10 and 6, has done nothing to make this limerick less appealing. It still has the regularity of meter, even when it lacks the magic of "special numbers".

Let's look at the claims Jones makes about ragtime music. The variety of syncopated rhythms in ragtime is such that I'd have to regard his Fibonacci match here as complete coincidence. The very catchy Pineapple Rag has groups of notes in patterns of 2,9,2,2 and 6. How much of a contortion is required here to fit an L-system? One of the best known Joplin rags, The Entertainer, is littered with patterns of 4s and 2s -- very un-Fib. There are patterns in the groupings but self-similarity across levels is a coincidence if it exists.

The essence of catchiness in music and rhythmic poetry is a balance of pattern and variation. There must be enough repetition of patterns to make your mind predict what will follow, and enough variation to entertain or surprise. By employing building blocks of pattern and variation at different scales or levels within a work (beat, phrase, theme, section) we produce something that is entertaining to the mind. Any self-similarity arises from applying our process of repetition+variation at different levels.

There are a lot of instances of "copying with errors" in nature, where a working pattern is repeated but with small variations. From the regularity of crystals (strongly patterned with very few defects or variations) to the variety of a forest (many trees of the same form but every one clearly unique), there is an abundance of examples of pattern+variation in nature. Is it an "inevitable reflection of nature" that causes us to find pattern+variation appealing in our entertainment, or is it mere coincidence? Unless we find alien intelligence from a universe where nature is predominately pattern-less, or dominated by patterns with almost no variation, we can never test that assertion. I think it's a long stretch to declare that our human preference for a balance of predictability and surprise arises from nature's collection of patterns. Nature's sheer variety of forms from the very regular to the unpredictable ensures that one can always find coincidental matches.


I know zero about limericks. I took a quick run through the article. Had similar problems/questions.

Ragtime is ragged time, march time played raggedly. We owe it to John Philip Sousa and the Sousa band phenomenon, which owes itself to the increase in band music in the Civil War and the weird interplay of brass instruments with African traditions. But to say that ragged time is called degenerate misses the point; it was the cacaphony, the dissonance and the blatant sexuality that was at issue, not the time. All good music breathes. Literally the hardest part in playing Chopin is the beautiful random quality of the time, as if the music were being created on the spot, impromptu. Empty Chopin is like "white jazz", played in straight time, without any sense of swing. Chopin's dance music swings. His friends noted that when he played his actual time signature shifted, moving from waltz time to half time, etc. slowing down and then speeding up. The degeneracy of dissonance - or then about atonality - is difficulty with the lack of resolution, when the harmonics combine in a rough fashion, when the music lacks a tonal center or when it fails to return to its "natural" end tones. Totally different thing. Excuse me if you know all this.

Yes, I've played enough Chopin to know what you mean, jk.

The more I go back and look at that essay, the less I like in it.

When Jones looks at the exact numbers of beats in each group and matches coincidences to Fibonacci numbers, I think he's missed the point completely. It's like saying "Here is an example of a world champion athlete. He parts his hair exactly down the centre. See how that symmetry makes him an ideal model of an athlete?"

Nature, music and mathematics do share a lot of patterns. A lot of music has symmetry and recurring patterns. There are beautiful parallels to be drawn between improvisation and natural selection, between ensemble jazz and co-evolution. Let's face it, though, the mathematics here is much harder to define and even harder to write in simple terms. It doesn't make for a simple essay, and it's harder to show point by point correlations between the music and the numbers.

Since Jones's stated interests are "exploring the underlying relationships between music, science and culture" he is bound to look for patterns common to the three, and seems prone to assignation of special significance to them when he finds them. Since simple elegant patterns are easier to recognize in very structured music and in rhythmic light verse, his stated interests bias him toward a simple extension of the questionable foundations of western classicism: that perfection can be found in clarity, concision, balance and structure. By substituting the mathematics of recusive structures for the simple geometry of classicism, he's still locked himself to a classical idea that there exists a Grand Unifying Theory of mathematics and music.

I feel his predisposition to recognize patterns and highlight them is far more religious than scientific. To select just a few cases where he's discovered a pattern that ties verse, music and maths together, but ignore the wealth of counter examples, is the stuff of conspiracy theorists. A more intellectually honest thesis to take would be:
1. L-systems create interesting, layered, self-similar patterns.
2. There are some pleasing musical forms that can be modeled using L-systems.
3. Therefore L-systems have the potential to be used to help generate pleasing music.

If he's so certain there's an underlying pattern he could have even tried to model all Learian limericks with L-systems. If he could show that the Limericks most pleasing to an independent audience were also the ones that were flawlessly modeled by a simple system, he'd have at least a shred of evidence for his GUT assumption.

Instead he insists without support that "during the Twentieth Century the Fibonacci paradigm has emerged triumphant, and entered the mainstream to form a more subtle and natural structural template, or cultural point of reference, against which our minds can measure their aesthetic response."

What next? A return to the Pythagorean "music of the spheres" as the basis for understanding music and mathematics?